Tri :
Date
Editeur
Auteur
Titre


Andras Vasy  Microlocal analysis and wave propagation (Part 2)
/ Fanny Bastien
/ Canalu.fr
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In these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on manifolds without boundary, and on manifolds with boundary. In the latter case there is no `standard' algebra of differential, or pseudodifferential, operators; I will discuss two important frameworks: Melrose's totally characteristic, or b, operators and scattering operators. Apart from the algebraic and mapping properties, I will discuss microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as normally hyperbolic trapping. The applications discussed will include Fredholm frameworks (which are thus global even for non elliptic problems!) for the Laplacian on asymptotically hyperbolic spaces and the wave operator on asymptotically de Sitter spaces, scattering theory for `scattering metrics' (such as the `large ends' of cones), wave propagation on asymptotically Minkowski spaces and generalizations (`Lorentzian scattering metrics') and on Kerr de Sitter type spaces. The lectures concentrate on linear PDE, but time permitting I will briefly discuss nonlinear versions. The lecture by the speaker in the final workshop will use these results to solve quasilinear wave equations globally, including describing the asymptotic behavior of solutions, on Kerr de Sitter spaces. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Maciej Zworski  From redshift effect to classical dynamics : microlocal proof of Smale's conjecture
/ Fanny Bastien
/ 30062014
/ Canalu.fr
Zworski Maciej
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Dynamical zeta functions of Selberg, Smale and Ruelle are analogous to the Riemann zeta function with the product over primes replaced by products over closed orbits of Anosov flows. In 1967 Smale conjectured that these zeta functions should be meromorphic but admitted "that a positive answer would be a little shocking". Nevertheless the continuation was proved in 2012 by GiuliettiLiveraniPollicott. By combining the Faure Sjöstrand approach to Anosov flows and Melrose's microlocal radial estimates, Dyatlov and I gave a simple proof of that conjecture. The same radial estimates were used by Vasy to provide a microlocal explanation of the redshift effect and propagation estimates for Kerr de Sitter like spaces. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Andras Vasy  Quasilinear waves and trapping: Kerr‐de Sitter space
/ Fanny Bastien
/ 30062014
/ Canalu.fr
Vasy Andràs
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In this talk I will describe recent work with Peter Hintz on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally hyperbolic trapping results of Dyatlov and a Nash Moser iteration scheme. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Andras Vasy  Microlocal analysis and wave propagation (Part 1)
/ Fanny Bastien
/ 16062014
/ Canalu.fr
Vasy Andràs
Voir le résumé
Voir le résumé
In these lectures I will explain the basics of microlocal analysis, emphasizing non elliptic problems, such as wave propagation, both on manifolds without boundary, and on manifolds with boundary. In the latter case there is no `standard' algebra of differential, or pseudodifferential, operators; I will discuss two important frameworks: Melrose's totally characteristic, or b, operators and scattering operators. Apart from the algebraic and mapping properties, I will discuss microlocal ellipticity, real principal type propagation, radial points and generalizations, as well as normally hyperbolic trapping. The applications discussed will include Fredholm frameworks (which are thus global even for non elliptic problems!) for the Laplacian on asymptotically hyperbolic spaces and the wave operator on asymptotically de Sitter spaces, scattering theory for `scattering metrics' (such as the `large ends' of cones), wave propagation on asymptotically Minkowski spaces and generalizations (`Lorentzian scattering metrics') and on Kerr de Sitter type spaces. The lectures concentrate on linear PDE, but time permitting I will briefly discuss nonlinear versions. The lecture by the speaker in the final workshop will use these results to solve quasilinear wave equations globally, including describing the asymptotic behavior of solutions, on Kerr de Sitter spaces. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Jérémie Szeftel The resolution of the bounded L2 curvature conjecture in General Relativity (Part 4)
/ Fanny Bastien
/ 27062014
/ Canalu.fr
Szeftel Jérémie
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In order to control locally a space‐time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space‐like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well‐posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with Sergiu Klainerman and Igor Rodnianski. These lectures will start from scratch and require no specific background. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Jérémie Szeftel The resolution of the bounded L2 curvature conjecture in General Relativity (Part 1)
/ Fanny Bastien
/ 26062014
/ Canalu.fr
Szeftel Jérémie
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Voir le résumé
In order to control locally a space time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with Sergiu Klainerman and Igor Rodnianski. These lectures will start from scratch and require no specific background. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Jérémie Szeftel  General relativity (Workshop)
/ Fanny Bastien
/ 01072014
/ Canalu.fr
Szeftel Jérémie
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Voir le résumé
In order to control locally a space time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with Sergiu Klainerman and Igor Rodnianski. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Alexander Strohmaier  Workshop
/ Fanny Bastien
/ 03072014
/ Canalu.fr
Strohmaier Alexander
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I will explain how one can formulate and formalize the Gupta Bleuler framework for the Quantization of the electromagnetic field in an algebraic manner so that it works on globally hyperbolic space times. I will then discuss a construction of physical representations that works without the "spectral gap assumption" in the case of absense of zero energy resonances. These can be excluded by topologocial restrictions at infinity. This is based on joint work with Felix Finster. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Lionel Mason  Perturbative formulae for scattering of gravitational wave
/ Fanny Bastien
/ 03072014
/ Canalu.fr
Mason Lionel
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The Christodoulou Klainerman proof of existence of asymptotically simple spacetimes shows that it is reasonable to consider the scattering of characteristic data for the Einstein field equations from past null infinity to that on future null infinity in a neighbourhood of Minkowski space. In this talk I present new explicit perturbative formulae for this scattering for general data to arbitrary order. Unlike previous such formulae, these new formulae are not chiral, and naturally respect the real structures and may therefore be more amenable to analysis. This is based on joint work with Yvonne Geyer and Arthur Lipstein and with David Skinner. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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Philippe G LeFloch  Weakly regular spacetimes with T2 symmetry
/ Fanny Bastien
/ 01072014
/ Canalu.fr
Lefloch Philippe
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I will discuss the initial value problem for the Einstein equations and present results concerning the existence and asymptotic behavior of spacetimes, when the initial data set is assumed to be T2 symmetric and satisfies weak regularity conditions so that the spacetimes may exhibit impulsive gravitational waves and shock waves. This lecture is based on papers written over the period 2004—2014 and available at philippelefloch.org. Mot(s) clés libre(s) : mathématiques, Grenoble, école d'été, General Relativity, institut fourier, summer school, asymptotic analysis

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